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CHEBYSHEV INEQUALITIES WITH LAW-INVARIANT DEVIATION MEASURES

Published online by Cambridge University Press:  21 December 2009

Bogdan Grechuk ,
Anton Molyboha  and
Michael Zabarankin
Bogdan Grechuk
Affiliation:
Department of Mathematical Sciences, Stevens Institute of Technology, Castle Point on Hudson, Hoboken, NJ 07030 E-mail: bgrechuk@stevens.edu; amolyboh@stevens.edu; mzabaran@stevens.edu
Anton Molyboha
Affiliation:
Department of Mathematical Sciences, Stevens Institute of Technology, Castle Point on Hudson, Hoboken, NJ 07030 E-mail: bgrechuk@stevens.edu; amolyboh@stevens.edu; mzabaran@stevens.edu
Michael Zabarankin
Affiliation:
Department of Mathematical Sciences, Stevens Institute of Technology, Castle Point on Hudson, Hoboken, NJ 07030 E-mail: bgrechuk@stevens.edu; amolyboh@stevens.edu; mzabaran@stevens.edu
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Abstract

The consistency of law-invariant general deviation measures with concave ordering has been used to generalize the Rao–Blackwell theorem and to develop an approach for reducing minimization of law-invariant deviation measures to minimization of the measures on subsets of undominated random variables with respect to concave ordering. This approach has been applied for constructing the Chebyshev and Kolmogorov inequalities with law-invariant deviation measures—in particular with mean absolute deviation, lower semideviation and conditional value-at-risk deviation. Additionally, an advantage of the Kolmogorov inequality with certain deviation measures has been illustrated in estimating the probability of the exchange rate of two currencies to be within specified bounds.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 24 , Issue 1 , January 2010 , pp. 145 - 170
Copyright
Copyright © Cambridge University Press 2009

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